## The Logical Representation of Domainsf

Yixiang Chen*
Department of Mathematics
Shanghai Normal University
Shanghai 20034, P.R. China
ychen@shnu.edu.cn

Achim Jung
The School of Computer Science
The University of Birmingham
Edgbaston, Birmingham, B15 2TT
England
A.Jung@cs.bham.ac.uk

Abstract : The primary contribution of the presentation talk is to give a logical representation of L-domains found by the second author [3], based on disjunctive propositional theory introduced by the first author in his book [2], which comes from the work of Pitts' Horn propositional theory [5] and the idea of Johnstone with respect to disjunctive theories [4],which includes infinite disjunction, but subject to some restriction.

First, the authors gives the semantics of disjunctive proposition theory T in disjunctive semilattice (D-semilattice, for short) introduced by the first author [1], and then, the soundness and completeness of this logic system. A basic operator generating a D-semilattice A(T) from the theory T is given.

Secondly, the logical representation of stable dD-semialttice is given. Two kinds of disjunctive propositional theory T(L) and T*(L) generating from a stable D-semilattice L are given. Then, the authors show the following theorem.

Theorem A  Every stable dD-semilattice L is isomorphic with the induced D-semilattice A(T*(L)).

Finally, the authors investigate the logical representation of L-domains and Scott domains. In our investigation, we need the stable semitopology SN(D) for L-domain D, which is introduced by the first author in [1]. It was shown that the semitopology SN(D) for L-domain D is a stable dD-semilattice. So, we have two disjunctive proposition theory T(SN(D)) and T* (SN(D)). Our main results are the following theorems.

Theorem B  Given an L-domain D. Then D is isomorphic to the set consisting of models of a disjunctive proposition theory T * (SN(D)) into the two-point lattice q. That is ,

 D @ Mod(T* (SN(D)),q).

Theorem C  Given a Scott domain D. Then D is isomorphic to the set consisting of models of a disjunctive proposition theory T (SN(D)) into the two-point lattice q. That is ,

 D @ Mod(T (SN(D)),q).

Reference

[1] Yixiang Chen, Stone duality and representation of stable domains, Computers Math. Appl. Vol.34, No.1, pp27-41, 1997.

[2] Yixiang Chen. Stable domain theory of Formal Semantics(in Chinese). Science Press, Beijing, 2003.

[3] Achim Jung, Cartesian closed categories of algebraic CPOs, Theoret. Comput. Sci. Vol:70, pp 233-250, 1990.

[4] P.T. Johnstone, A syntactic approach to Diers' localizable category, In Applications of Sheaves (Proc. Res. Sympos. Appl. Sheaf Theory to Logic, Algebra and Anal., Durham, 1979; Lecture Notes in Mathematics, Vol.753, pp 466-478, Springer, Berlin, 1979.

[5] A. M. Pitts, Categorical logic, Manuscript, University of Cambridge Laboratory, Tech. Rept. No.367, May 1995. http:www.cl.cam.ac.uk/ amp12 .

f The corresponding author is supported by NSF of China(60273052), and EPSRC(GR/S79770/01), as well as the Key Project of the Educational Commission of Shanghai (02DZ46).

## On Meromorphic Functions with Finite Logarithmic Orderf

Peter Tien-Yu Chern
Department of Applied Mathematics
I-Shou University
Ta-Hsu Hsiang, Kaohsiung county
Taiwan 840, R.O.C.
tychern@isu.edu.tw

Abstract : By using a slow growth scale, the logarithmic order, with which to measure the growth of functions, we obtain basic results on the value distribution of a class of meromorphic functions of zero order.

2000 Mathematics Subject Classification. Primary 30D30, 30D35.

Key Words and Phrases: meromorphic function, value distribution, finite logarithmic order.

f This paper was supported in part by the NSC R.O.C. under the grant NSC 86-2115-M214-001, a fund from Academia Sinica, (Taipei, Taiwan), and a fund from Michigan State University, U.S.A.

## Continuous Solutions of a Singular Cauchy-Riemann System

Dao-Qing Dai
Department of Mathematics
Sun Yat-Sen(Zhongshan) University
Guangzhou 510275 China
stsddq@zsu.edu.cn

Abstract : We study solvability of the Riemann-Hilbert problem for a generalized Cauchy-Riemaim system with several singularities and reveal several new phenomenon. For the number of continuous solutions we shall show that it depends not only on the index but also on the location and type of the singularities, moreover it does not depend continuously on the location and type of the equation.

The investigation of the above problem is highly motivated by the fact that this model may serve to reveal difficulties that occur in generating a general theory for singular Vekua systems and in clarifying the situation in earlier attempts in the literature.

Also, open questions will be mentioned.

Reference

[1] I. N. Vekua, Stationary singularities of generalized analytic functions(Russian), Dokl. Akad. Nauk SSSR, Vol. 145(1962), 24-26.

[2] L. G. Mikhailov, A new class of singular equations and its application to differential equations with singular coefficients, Akademie-Verlag, Berlin, 1970.

[3] Z. D. Usmanov, Generalized Cauchy-Riemann systems with a singular point, Longman, Harlow, 1997.

[4] H. Begehr and D. Q. Dai, On continuous solutions of a generalized Cauchy-Riemann system with more than one singularity, Journal of Differential Equations, Vol. 196(2004), 67-90.

## Differential Subordination Relations of Some Classes of Analytic Functions

Chunyi Gao* and Shaomou Yuan
College of Mathematics and Computing Science,
Changsha University of Science & Technology
Changsha, Hunan 410076, P. R. China
gaochy@csust.edu.cn,  shaomouyuan@163.com

Abstract : In this paper , three new classes of analytic functions are introduced by using Hadamard Product. These three classes are denoted by Tgr(p  ; A  ,  B )  , Rgr(p  ; A  ,  B ) and Qgr(p  ; A  ,  B). By studying these classes , we respectively obtain some new subordination relations of them . Our results generalize some known work . The main conclusions of this paper are:

Theorem 1. Let f(z) Î Tgr(p  ; A1(r/p)  ,  B1(r/p)), r ³ 0, A and B be such that -1 < B < A £ 1, then [(f*g)(z)]/zp \prec h(A,B;z)   (|z| < 1), where A1(r) and B1(r) are respectively defined as follows :

 A1(r) = (A-B)[r(1-B2)+(1+|B|)2]2+B(1-AB)(1+|B|)4 (1+|B|)2(1-B2)[r(1-B2)+(1+|B|)2]
 B1(r) = B(1+|B|)2 r(1-B2)+(1+|B|)2 ,
and h(A,B;z) = (1+Az)/(1+Bz) .

Theorem 2. Let f(z) Î Rgr(p  ; A2(r)  ,  B2(r)), r ³ 0, A and B satisfy -1 < B < A £ 1, then [(f*g)(z)]/f(z)\prec h(A,B;z) (|z| < 1), where A2(r) and B2(r) are respectively defined as follows :

 A2(r) = rf(A,B)y(A,B)+ A-B 1-B2 + B(1-AB)(A-B) r(1-B2)2f(A,B)y(A,B)+(1-B2)(A-B)
 B2(r) = B(A-B) r(1-B2)f(A,B)y(A,B)+A-B ,
and
f(A,B) = (A-B)(1+B)
(1+A)(1+|B|)2
,   y(A,B) =
 __________Ö(1-A2)(1-B2)

1-AB)
.

Theorem 3. Let f(z) Î Qgr(p  ; A2(r)  , B2(r)), r ³ 0, A and B satisfy -1 < B < A £ 1, then [(f*g)(z)]/zp\prec h(A,B;z)  (|z| < 1) , where A2(r) and B2(r) are defined as above.

## Rational Solutions of Algebraic Differential Equations

Guoting Chen
Laboratoire P. Painlevé,
Université de Lille 1,
59655 Villeneuve d'Ascq, France
Guoting.Chen@math.univ-lille1.fr

Yujie Ma*
Key Lab of Mathematics Mechanization
Academy of Mathematics and Systems Science
Beijing 100080, China
yjma@mmrc.iss.ac.cn

Abstract : Rational solutions of algebraic differential equations are considered. We give an effective necessary condition for an algebraic differential equation of first order or second order which admits rational function as its general solution. We also present an effective algorithm to decide whether a given algebraic differential equation of first order or second order admits rational general solution and if so compute it. Some results on the rational solutions of a first order algebraic differential equation are presented if its general solution is not a rational function.

## Some Properties of the Solutions to Generalized Modular Equations

Songliang Qiu* and X.Y. Ma
Department of Mathematics
Hangzhou Dianzi University
Hangzhou 310018, P.R. China
sl_qiu@hziee.edu.cn

Abstract : In 1995, B. Berndt, S. Bhargava and F. Garvan published an important paper [1], in which they studied generalized modular equations and gave proofs for numerous statements concerning these equations made by S. Ramanujan in his unpublished notebooks. Since then some important properties of the solutions to the generalized modular equations, which are sometimes called the modular functions with signature 1/a and degree p for 0 < a < 1 and p > 0, have been obtained. In this paper, the authors study the monotonicity and concavity properties of certain combinations in terms of the modular function, and show several significant results for this function, including its sharp bounds. In addition, some properties of the generalized Grötzsch ring function and generalized elliptic integrals, which are related to the modular function, are presented.

## Einstein-Kähler Metric on Cartan-Hartogs Domains

An Wang*
Department of Mathematics
Capital Normal University
Beijing 100037, China
wangancn@sina.com

Weiping Yinf
Department of Mathematics
Capital Normal University
Beijing 100037, China
wyin@mail.cnu.edu.cn or wpyin@263.net

Keywords: Cartan-Hartogs domain, Einstein-Kähler metric, Holomorphic sectional curvature, Generating function.

MR(2000) 32H02

1. Introduction

Cheng and Yau[CY] proved that any C2 bounded pseudoconvex domain W in Cn has a complete Einstein-Kähler metric. Without any regularity assumption on the domain W, Mok and Yau[MY] proved that the complete Einstein-Kähler metric always exists. This Einstein-Kähler metric is given by

EW(z): = å
2g
 ¶zi ¶ z j
dzi
dzj

,
where g is a unique solution to the boundary problem of the Monge-Amp\` ere equation:
ì
ï
ï
í
ï
ï
î
 det æç è ¶2g ¶zi ¶[`z]j ö÷ ø = e(n+1)g
 z Î W
 g = ¥
 z Î ¶W.
We call g the generating function of EW(z). It is obvious that if one determines g explicitly, then the Einstein-Kähler metric is also determined explicitly.

The explicit formulas for the Einstein-Kähler metric, however, are only known on homogeneous domains. In the paper [WU], H.Wu points out that among the four classical invariant metrics(i.e. the Bergman metric, Carathéodory metric, Kobayashi metric and Einstein-Kähler metric), the Einstein-Kähler metric is the hardest to compute because its existence is proved by complicated nonconstructive methods. The purpose of this paper is to compute the explicit formulas of complete Einstein-Kähler metrics on Cartan-Hartogs domains of the following four types:

 YI(1,m,n;K): =
 { w Î C,Z Î RI(m,n):|w|2K < det (I-Z Z T ),K > 0}: = YI,
 YI I(1,p;K): =
 {w Î C,Z Î RI I(p):|w|2K < det (I-Z Z T ) ,K > 0}: = YI I,
 YI I I( 1,q;K): =
 {w Î C,Z Î RI I I(q): |w|2K < det (I-Z Z T ) ,K > 0}: = YI I I,
 YI V(1,n;K): =
 {w Î C,Z Î RI V(n) : |w|2K < 1-2Z Z T +|ZZT|2,K > 0,}: = YI V.
Where RI(m,n), RII(p), RI I I(q) and RI V(n) are the first, second, third and fourth Cartan domains respectively in the sense of Loo-Keng HUA [Hu], [`Z]T indicates the conjugate and transpose of Z, det indicates the determinant. The Cartan-Hartogs domains are introduced in 1998. The Bergman kernel functions on Cartan-Hartogs domains are obtained in explicit formulas in [Yin1,Yin2,Yin3,GY]. And these Bergman kernel functions are Bergman exhaustions, therefore all of the Cartan-Hartogs domains are bounded pseudoconvex domains. Some results on Hua domains can be found in [Yin1-Yin3,GY,YW,YWZ]. With some obvious exceptions the Cartan-Hartogs domains are non-homogeneous.

* Supported by National Natural Science Foundation of China (Grant No. 10071051) and Natural Science Foundation of Beijing (Grant No. 1002004) and Science and Technology Development Foundation of Beijing Education

f Supported by National Natural Science Foundation of China (Grant No. 10171068) and Natural Science Foundation of Beijing (Grant No. 1012004)

## A Note about Bruck Conjecture and Its Applications on Uniqueness Theory of Meromorphic Functions

Jun Wang
Institute of Mathematics
Fudan University
Shanghai 200433, P.R. China
wangjun-rose@sohu.com

Abstract : This lecture is devoted to studying the uniqueness problem of entire functions that share a small function with their derivatives. There are three theorems of such type, generalizing related previous results by several authors.

## Some Positive Answers to Serre Problem

Jinhao Zhang
Department of Mathematics
Fudan University
Shanghai 200433, P.R. China
seminar@fudan.edu.cn

Abstract : In 1953, Serre raised the problem: If E is a holomorphic fiber bundle with a Stein base and a Stein fiber, is E Stein? There are many mathematicians have made contributions to this problem. In this talk I will show a criterion which unifies some known cases and gives applications to resolving Serre's problem in the affirmative under some new analytic or geometric assumptions.

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On 08 Dec 2004, 10:37.